Why do we use the term multicollinearity, when the vectors representing two variables are never truly collinear?

Question

When two vectors $a$ and $b$ are collinear, then $a = xb$, (where $x$ is a scalar) so in linear algebra, collinearity is a narrowly and clearly defined (and binary) concept. Two vectors -- in my understanding -- are either collinear or they are not; there are no different degrees of collinearity. If $a$ and $b$ represent distinct time series or samples in econometrics, I would therefore expect to never actually find (multi)collinearity in any empirical context, because it is close to impossible that two distinct time series or samples are exact multiples of one another. We can find correlation, and I could also imagine finding variables which are -- at least approximately -- coplanar in certain contexts, but never true collinearity.

Why then is the term multicollinearity used in the context of econometrics, when what it really seems to mean is a (statistically significant) correlation between two or more explanatory variables in a multiple regression model? Is what's meant by problems of multicollinearity that the null hypothesis of no collinearity is rejected at a certain significance level, even when we know for certain that there is strictly speaking no collinearity among the explanatory variables simply by inspecting the data? Why is multiple correlation not the more accurate term to use?

I have recently encountered the terms perfect and imperfect multicollinearity and this has confused me additionally. Does someone have a rigorous understanding of this and could share it? I would very much appreciate it!

Answer 1

I don't think anyone worries about exact collinearity. If that was the case, $X'X$ would not be invertible. That is why full column rank of $X$ in one of the first assumptions. People worry about inexact relationships, since then there are coefficients to interpret, but they are too unreliable to be useful. The world multicollinearity is usually reserved for the latter. But one can easily become the other in the limit. Think about $\vert X'X\vert$. This determinant declines in value with increasing collinearity, tending to zero as the collinearity becomes exact. You can also consider the auxiliary regression $R^2_j$ tending to 1. Perhaps that is why we use the same term for both, though the extreme case is impossible if you have coefficients in hand.

Exact linear relationships most often arise in the context of the dummy variable trap and are easy to diagnose. The consequences of approximate linear relationship among some of the regressors in the sample at hand are indistinguishable from the consequences of inadequate variability of the regressors in a data set. Arthur Golberger joked that we should call this phenomenon "micronumerosity" instead. This is the one that people usually worry about, though it doesn't violate any of our usual assumptions.

This sort of multicollinearity, by definition, is a feature of your particular sample of data with which you're trying to fit a regression model. More or less, it means that there is insufficient information in your data to make reliable inferences about the individual parameters of the underlying (population) model, though jointly they may well be informative.

There are various sample measures that you can compute and report (VIFs or conditioning indices), to help gauge how severe this problem may be. But they're not statistical tests. Because multicollinearity is a characteristic of the sample, and not a characteristic of the population, you cannot test for it, in the same sense that it makes no sense to test that $\hat \beta = 0$, rather than $\beta = 0$. Of course, there are tests for relationships in the population that are often misinterpreted as test of multicollinearity.